Nonparametric Bootstrap Confidence Intervals for High-Quantiles of Heavy-Tailed Distributions

In risk management areas such as reinsurance, the need often arises to construct a confidence interval for a quantile in the tail of the distribution; for example, there is high probability that the sample maximum lies near or below the quantile. While different methods, including subsample bootstrap, are available for this purpose, a full-sample bootstrap has not been shown to be valid. In fact, doubts have been raised about the validity of full-sample bootstrap. In this paper, we show that the full-sample bootstrap is valid for constructing a confidence interval for a high-quantile of heavy-tailed distribution. This is a significant theoretical contribution to the topic, and opens the possibility of full-sample bootstrap for inference on tails in higher dimensions. In our simulation study, our bootstrap method performed overall better than the data tilting method, but none was uniformly the best; the data tilting method appears to be currently the preferred choice. Since the two methods are based on quite different approaches but performed competitively in our simulation, we recommend that both methods be used side by side in applications